1

Product Mix Example

Static Workforce Planning models

Blending Problem

Aggregate Production Planning

Reference:

Winston and Albright, Chapters 2 & 3

Modeling and Solving LPs by Excel Solver

2

Product Mix Problem

Monet company makes four types of frames.

The following table gives the manufacturing data:

How many frames of each type should be made per week to maximize the profit?

Labor Metal Glass Profit Upperbound

Frame 1 2 4 6 $6 1000

Frame 2 1 2 2 $2 2000

Frame 3 3 1 1 $4 500

Frame 4 2 2 2 $3 1000

Resourcelimits perweek

4000 6000 10000

3

Product Mix Problem (contd.)

Linear Programming Formulation:

Max z = 6x1 + 2x2 + 4x3 + 3x4

subject to

6x1 + 2x2 + 4x3 + 3x4 4000 (labor)

4x1 + 2x2 + x3 + 2x4 6000 (metal)

6x1 + 2x2 + x3 + 2x4 10000 (glass)

x1 1000x2 2000x3 500x4 1000

x1, x2, x3 , x4 0

4

Product Mix Problem (contd.)

Product Mix Problem with Optimal Solution

Input dataFrame Type

1 2 3 4 Total used Total availableLabor hours per frame 2 1 3 2 4000 <= 4000Metal (oz.) per frame 4 2 1 2 6000 <= 6000Glass (oz.) per frame 6 2 1 2 8000 <= 10000

Total profit

Profit per frame $6.00 $2.00 $4.00 $3.00 $9,200.00

Production planFrame Type

1 2 3 4

Frames produced 1000 800 400 0

<= <= <= <=Maximum sales 1000 2000 500 1000

5

Answer & Sensitivity ReportsTarget Cell (Max)

Cell Name Original Value Final Value$F$10 Total profit $9,200.00 $9,200.00

Adjustable CellsCell Name Original Value Final Value

$B$15 Type 1 frames produced 1000 1000$C$15 Type 2 frames produced 800 800$D$15 Type 3 frames produced 400 400$E$15 Type 4 frames produced 0 0

ConstraintsCell Name Cell Value Formula Status Slack

$F$6 Labor hours 4000 $F$6<=$H$6 Binding 0$F$7 Metal (oz.) 6000 $F$7<=$H$7 Binding 0$F$8 Glass (oz.) 8000 $F$8<=$H$8 Not Binding 2000$B$15 Type 1 frames nonnegativity 1000 $B$15>=0 Not Binding 1000$C$15 Type 2 frames nonnegativity 800 $C$15>=0 Not Binding 800$D$15 Type 3 frames nonnegativity 400 $D$15>=0 Not Binding 400$E$15 Type 4 frames nonnegativity 0 $E$15>=0 Binding 0$B$15 Maximum type 1 frames 1000 $B$15<=$B$17 Binding 0$C$15 Maximum type 2 frames 800 $C$15<=$C$17 Not Binding 1200$D$15 Maximum type 3 frames 400 $D$15<=$D$17 Not Binding 100$E$15 Maximum type 4 frames 0 $E$15<=$E$17 Not Binding 1000

6

Answer & Sensitivity Reports

Changing CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$B$15 Type 1 frames produced1000 2 6 1E+30 2$C$15 Type 2 frames produced800 0 2 6 0.25$D$15 Type 3 frames produced400 0 4 2 0.5$E$15 Type 4 frames produced 0 -0.2 3 0.2 1E+30

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$F$6 Labor hours 4000 1.2 4000 250 1000$F$7 Metal (oz.) 6000 0.4 6000 2000 500

7

Static Work Scheduling Problem

Number of full time employees on different days of the week

Each employee must work five consecutive days and then receive two days off

Meet the requirements by minimizing the total number of full time employees

Day 1 = Monday 17

Day 2 = Tues. 13

Day 3 = Wedn. 15

Day 4 = Thurs. 19

Day 5 = Friday 14

Day 6 = Satur. 16

Day 7 = Sunday 11

8

Static Workforce Scheduling

LP Formulation:

Min. z = x1+ x2 + x3 + x4 + x5 + x6 + x7

subject to

x1 + x4 + x5 + x6 + x7 17x1+ x2 + x5 + x6 + x7 13x1+ x2 + x3 + x6 + x7 15x1+ x2 + x3 + x4 + x7 19x1+ x2 + x3 + x4 + x5 14 x2 + x3 + x4 + x5 + x6 16 x3 + x4 + x5 + x6 + x7 11

x1, x2, x3, x4, x5, x6, x7 0

9

Static Workforce Scheduling (contd.)

Post Office Scheduling Problem

Starting day of 5-day shift

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Total employee

sNumber starting 6.33 3.33 2.00 7.33 0.00 3.33 0.00 22.33

Number working on each

day 17.00 13.00 15.00 19.00 19.00 16.00 12.67>= >= >= >= >= >= >=

Minimal number required

each day 17 13 15 19 14 16 11

10

Static Workforce Scheduling (contd.)

Post Office Scheduling Problem with Integer Constraints

Starting day of 5-day shift

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Total employee

sNumber starting 7 4 2 8 0 2 0 23

Number working on each

day 17 13 15 21 21 16 12>= >= >= >= >= >= >=

Minimal number required

each day 17 13 15 19 14 16 11

11

Static Workforce Scheduling (contd.)

Generally, such problems have multiple optimal solution. We may be interested in an optimal solution which maximizes the number of days with weekends off. How to obtain such a solution?

How to create a fair schedule for employees so that all employees get weekends off?

We assumed demands are static with time. What if demands are functions of time? Such problems are called dynamic workforce scheduling problems.

How to allocate overtimes, and allow part-time employees?

12

Blending Problems

Situations where various inputs must be blended in some desirable proportion to produce goods for sale are called blending problems.

Examples of blending problems:

Blending various crude oils to produce different types of gasoline

Blending various types of metal alloys to various types of steels

Blending various livestock feeds to produce minimum-cost feed mixture for cattle

Mixing various types of paper to produce recycled paper of varying quality

13

Blending Problems (contd.)

Chandler Oil manufactures gasoline and heating oil

These products are produced by blending two types of crude oil (crude 1, and crude 2)

The following table gives the data for quality points and sales and purchase prices:

Determine the production quantities of gasolines and heating oils to maximize the profit

Sellingprice perbarrel

Minimumqualitypoints

Advertizingcost perbarrel

Gasoline $25.00 8 $0.20

Heating Oil $20.00 6 $0.10

Cost perbarrel

Qualitypoints

Availabilityin barrels

Crude 1 $25.00 10 5000

Crude 2 $20.00 5 10000

14

Blending Problem (contd.)

Decision Variables:

x11 : Number of barrels of crude 1 used in the manufacturing of gasoline

x12 : Number of barrels of crude 1 used in the manufacturing of heating oil

x21 : Number of barrels of crude 2 used in the manufacturing of gasoline

x22 : Number of barrels of crude 2 used in the manufacturing of heating oil crude 2

15

Blending Problem (contd.)

Max. (25 - .02) (x11 + x21) + (20 - .01) (x12 + x22)

s. t.x11 + x12 5000x21 + x22 10000

10x11 + 5x21 8(x11 + x21) 10x12 + 5x22 6(x12 + x22)

x11, x12 , x21, x22 0

16

Blending Problem (contd.)Chandler Blending Problem (nonoptimal solution)

Monetary inputs Gasoline Heating oilSelling price/barrel $25.00 $20.00

Advertising cost/barrel $0.20 $0.10

Quality level per barrel of crudes

Crude oil 1 10Crude oil 2 5

Required quality level per barrel of productGasoline Heating oil

8 6

Blending plan (barrels of crudes in each product)Gasoline Heating oil Barrels used Barrels available

Crude oil 1 0 3000 3000 <= 5000Crude oil 2 5000 0 5000 <= 10000

Barrels sold 5000 3000

Constraints on qualityGasoline Heating oil

Quality "points" obtained 25000 30000>= >=

Quality "points" required 40000 18000

Profit summaryGasoline Heating oil

Profit/barrel $24.80 $19.90

Total profit $183,700

17

Blending Problem (contd.)Cell Name Value Cost Coefficient Increase Decrease$B$17 Crude oil 1 Gasoline 3000 0 24.8 58.1666667 8.166666667$C$17 Crude oil 1 Heating oil 2000 0 19.9 8.16666667 58.16666667$B$18 Crude oil 2 Gasoline 2000 0 24.8 87.25 6.125$C$18 Crude oil 2 Heating oil 8000 0 19.9 6.125 14.54166667

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$D$17 Crude oil 1 Barrels used5000 29.7 5000 10000 2500$D$18 Crude oil 2 Barrels used10000 17.45 10000 10000 6666.666667$B$23 Quality points obtained Gasoline40000 -2.45 0 5000 20000$C$23 Quality points obtained Heating oil60000 -2.45 0 10000 6666.666667

18

Blending Problem (contd.)

MODELING ISSUES:

Blending problems in practice have many more inputs and outputs

Quality level of gasoline and heating oil may not a linear function of the fractions of the inputs used

Blending problems are periodically solved on the basis of current inventories and demand forecasts